Another well-known result of Gromov that I am quite fond of is his nonsqueezing theorem in symplectic geometry (or Hamiltonian mechanics). In its original form, the theorem states that a ball of radius R in a symplectic vector space (with the usual symplectic structure ) cannot be mapped by a symplectomorphism into any cylinder which is narrower than the ball (i.e. ). This result, which was one of the foundational results in the modern theory of symplectic invariants, is sometimes referred to as the “principle of the symplectic camel”, as it has the amusing corollary that a large “camel” (or more precisely, a 2n-dimensional ball of radius R in phase space) cannot be deformed via canonical transformations to pass through a small “needle” (or more precisely through a 2n-1-dimensional ball of radius less than R in a hyperplane). It shows that Liouville’s theorem on the volume preservation of symplectomorphisms is not the only obstruction to mapping one object symplectically to another.

I can sketch Gromov’s original proof of the non-squeezing theorem here. The symplectic space can be identified with the complex space , and in particular gives an almost complex structure J on the ball (roughly speaking, J allows one to multiply tangent vectors v by complex numbers, and in particular Jv can be viewed as v multiplied by the unit imaginary i). This almost complex structure J is compatible with the symplectic form ; in particular J is tamed by , which basically means that for all non-zero tangent vectors v.

Now suppose for contradiction that there is a symplectic embedding from the ball to a smaller cylinder. Then we can push forward the almost complex structure J on the ball to give an almost complex structure on the image . This new structure is still tamed by the symplectic form on this image.

Just as complex structures can be used to define holomorphic functions, almost complex structures can be used to define pseudo-holomorphic or J-holomorphic curves. These are curves of one complex dimension (i.e. two real dimensions, that is to say a surface) which obey the analogue of the Cauchy-Riemann equations in the almost complex setting (i.e. the tangent space of the curve is preserved by J). The theory of such curves was pioneered by Gromov in the paper where the nonsqueezing theorem was proved. When J is the standard almost complex structure on , pseudoholomorphic curves coincide with holomorphic curves. Among other things, such curves are minimal surfaces (for much the same reason that holomorphic functions are harmonic), and their symplectic areas and surface areas coincide.

Now, the point lies in the cylinder and in particular lies in a disk of symplectic area spanning this cylinder. This disk will not be pseudo-holomorphic in general, but it turns out that it can be deformed to obtain a pseudo-holomorphic disk spanning passing through of symplectic area at most . Pulling this back by , we obtain a minimal surface spanning passing through the origin that has surface area at most . However, any minimal surface spanning and passing through the origin is known to have area at least , giving the desired contradiction. [This latter fact, incidentally, is quite a fun fact to prove; the key point is to first show that any closed loop of length strictly less than in the sphere must lie inside an open hemisphere, and so cannot be the boundary of any minimal surface spanning the unit ball and containing the origin. Thus, the symplectic camel theorem ultimately comes down to the fact that one cannot pass a unit ball through a loop of string of length less than .]

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